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A117618
Least number with complexity height of n, under integer complexity A005245.
3
1, 6, 7, 10, 22, 683
OFFSET
1,2
COMMENTS
Consider the recursion: A005245(n), A005245(A005245(n)), A005245(A005245(A005245(n))), ... which we know is finite before reaching a fixed point, as A005245(n) <= n. The number of steps needed to reach such a fixed point is the complexity height of n (with respect to the A005245 measure of complexity, there being others in the OEIS).
a(7) >= 872573642639 = A005520(89). - David A. Corneth, May 06 2024
REFERENCES
W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971.
R. K. Guy, Unsolved Problems in Number Theory, Sect. F26.
LINKS
W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy]
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
E. Pegg, Jr., Integer Complexity.
Eric Weisstein's World of Mathematics, Integer Complexity.
FORMULA
a(n) = least k such that A005245^(n)(k) = A005245^(n-1)(k) but (if n>1) A005245^(n-1)(k) != A005245^(n-2)(k), where ^ denotes repeated application.
For n >= 3, a(n) = A005520(a(n-1)). - Max Alekseyev, May 06 2024
EXAMPLE
a(1) = 1 because the A005245 complexity of 1 is 1, already giving a fixed point.
a(2) = 6 because it is the smallest x such that A005245(x) =/= x and A005245(x) = A005245(A005245(x)).
a(3) = 7 because 7 is the least number x with complexity 6, thus taking a further step of recursion to reach a fixed point.
a(4) = 10 because 10 is the least number with complexity 7.
a(5) = 22 because 22 is the least number with complexity 10.
a(6) = 683 because 683 is the least number with complexity 22.
a(7) = the least number with complexity 683.
KEYWORD
nonn,more,hard,changed
AUTHOR
Jonathan Vos Post, Apr 07 2006
EXTENSIONS
a(2)=6 inserted by Giovanni Resta, Jun 15 2016
Edited by Max Alekseyev, May 06 2024
STATUS
approved